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GENERAL FORMULATION OF A PERTURBATION

THEORY FOR UNSTEADY CAVITY FLOWS

BY

D. P. Wang and T. Yao-tsu Wu

Hydrodynamics Laboratory

~ a / r r n i n Laboratory of Fluid Mechanics and J e t Propulsion

California Institute of Technology

Pasadena, California

Report NO. 97.7 March 1965

ABSTRACT

The problem of a two-dimensional cavity flow of an ideal fluid

with smal l unsteady disturbances in a gravity f ree field i s considered.

By regarding the unsteady motion a s a smal l perturbation of an estab-

lished steady cavity flow, a fundamental formulation of the problem i s

presented. It i s shown that the unsteady disturbance generates a surface

wave propagating downstream along the f ree cavity boundary, much in

the same way a s the classical gravity waves in water, only with the cen-

trifugal acceleration owing to the curvature of the streamlines in the

basic flow playing the role of an equivalent gravity effect. As a particu-

la r ly simple example, the surface waves in a hollow potential vortex

flow i s calculated by using the present theory.

INTRODUCTION 1

(2 Since the pioneering contributions of ~e lmhol tz ' ' ) and Kir chhoff ,

nearly a century ago, on the subject of steady, irrotational, plane flows

involving f ree streamlines, extensive applications have been made to jets

and to flows with a cavity or wake formation. In spite of such a long his-

- tory and the mature state of steady f ree streamline theory and i ts wide ap-

plications to engineering problems, the subject of unsteady cavity flows

has received attention only in the last seventeen years or so. Some of the

difficulties involved in unsteady cavity flows can be envisaged a s follows.

The theoretical treatment of ir r otational, two -dimensional cavitating flows of

an ideal fluid i s usually based on a certain proposed physical model, for ex-

ample, the Kirchhoff-Helniholtz model. If the flow i s steady, the exact

solution of such a problem, within the assumption of the proposed model, i s

usually obtained by using the hodograph method, since in this case a sur - face of constant pressure i s also one of constant speed. This property,

however, no longer holds valid in the case when the flow i s unsteady. Con-

sequently, in order to investigate some of the characteristics of unsteady

cavitating flows, different approaches and approximations have been intro-

duced by various authors. Some of the early contributions have been dis-

cussed by ~ i l b a r & ~ ) , Birkhoff and ~ a r a n t o n e l l o ' ~ ) . In order to help ap-

praise the present state of the knowledge, a brief survey may be made here

of the recent developments.

Numbers in parenthesis refer to similarly numbered references in bibliography a t end of paper.

In 1949 von ~ 6 r m & n ( ~ ) t reated a n accelerated flow normal to a

flat plate held fixed in an inertial f r ame such that with a certain a c -

celeration, the flow separates f rom the plate to form a closed cavity

of constant shape attached behind the plate, and he obtained a solution

fo r a particular Froude number characterizing the acceleration. The

entire se t of flows for cavities with constant shape was later derived by

~ i l b a r ~ ( ~ ) for a rb i t r a ry polygonal obstacles. Fo r cavities with varying

shape, ~ i l b a r ~ ( ~ ) proposed the assumption that the f ree boundary, which

i s a mater ia l line, may be approximated by a streamline. As pointed

out by Gilbarg, it seems physically reasonable that e r r o r s f rom this

approximation may be quite small , a t leas t for not too rapidly varying

flows. Adopting this approximation, t reated the unsteady

cavitating flow pas t curved obstacles with a finite cavity closed in the

r e a r by a second fictitious body, a s in the Riabouchinsky model for

steady cavity flows(3 ). Noticing the es sential difference between the

two distinct cases when the fluid a t infinity i s accelerating or when the

body i s accelerating (in an inert ial f r ame) , ~ i h ( ~ ) t r e a t e d both cases , de-

riving general formulae for unsteady cavity flows when the velocity po-

tential Q assumes the fo rm Q = U(t)f(x, y).

F o r the general case of unsteady cavity flows, another approach

i s to regard the unsteady par t of the motion a s a smal l perturbation of

a steady cavity flow already established. With this approach Ablow and

Hayes ( 9 ) developed a perturbation theory which was later employed by

Fox and Morgan to investigate the stability problem of some f r e e sur -

face flows. Also, ~ u r l e " ' ) considered the large and small time solu-

tions of a jet issuing f rom a slit. In these perturbation theories, the

exact, linearized boundary conditions on the f ree surface a r e used. In

this category a somewhat different perturbation theory has been applied

to severa l specific problems by ~ o o d s " ~ ) , I?arkin(13), wu(14), Timman (15)

and Geur s t ( I6 ' 17) . In these lat ter works the f ree surface of an unsteady

cavity flow i s approximated by a s t r eamline, thus releasing completely

the kinematic condition imposed on the f r e e boundary. By doing so , it i s

hoped that such approximation can give satisfactory resul ts , perhaps for

slowly varying flows. Based on such an approximation the resulting flow

has been interpreted(12) to contain the effect that an unsteady disturbance

applied on the solid body will produce two vortex sheets leaving the separa-

tion points, propagating downstream on the f ree surface of the cavity with

a velocity equal to that

grounds i t can perhaps

approximation that the

of the f r e e s t r e am of the basic flow. On physical

be argued that the linearized theory based on the

mater ia l lines be replaced by streamlines would

become l e s s consistent and l ess accurate for moderately and rapidly

varying flows. On the other hand, the approach of Ablow and Hayes

seems to have not yet been fully extended to t r ea t the general case of un-

steady cavity flows. It i s the purpose of the present work to present a

consistent formulation of a perturbation theory for the general case,

following a method ra ther independent of that of Ablow and Hayes.

By assuming the time-dependent pa r t of the flow to be small , a

perturbation theory i s developed here by a systematic linearization in

the physical plane, without assuming that the displaced f r ee surface of the

cavity be approximated by a streamline. F r o m this general formula-

tion i t i s seen that the unsteady motion of the solid body produces in

gener-a1 f ree surface dynamic waves propagating along the cavity

boundary, much the same a s the gravity waves generated by a float-

ing body in motion. The centrifugal force owing to the curvature of the

s treamlines in the basic flow now plays the role of a n equivalent gravity

in the classical water wave problem. In this sense, the unsteady cavity

flows a r e s imilar in nature to the radiation of gravity waves over a

flat water surface, only now in a much more complex form since the

centrifugal acceleration var ies along the cavity surface. Such a dynamic

wave phenomenon cannot be found in the theory using the streamline -

approximation mentioned previously. A simple illustration of the pr e -

sent formulation i s ca r r i ed out for the surface waves over a hollow

vortex fir s t t reated by Lord ~ e l v i n ' ' ~ ) . Numerical resul ts of typical

unsteady cavity flows by using the present theory generally involve ex-

tensive analytical details ; such resul ts will be presented in a later work.

It is the hope of this paper to stimulate further interest in developing

this important and interesting subject, and in making applications for

engineering purposes.

GENERAL THEORY

W e suppose that for the time t < O a steady, irrotational, two-

almensioslal flow pas t a solid body has been established (in a gravity-

f ree field), i t s solution being assumed to be known. For t > O the solid

bodv i s given a n unsteady smal l disturbance, whose magnitude i s char-

acterized by a small parameter E . The resulting flow will be assumed

to remain irrotational in a region containing the body-cavity system.

We shall establish a perturbation theory, to the f i r s t order in E , by

regarding the time'-dependent par t of the flow a s a small perturbation

of the basic steady flow.

Under this assumption the flow possesses a velocity potential

~ ( x , y, t; E ) which may be expanded for t > O a s

P(x, y, t ; E ) = (P ( x Y y ) t E (4 (x, yy t ) t 0(E2 ) * 0 (1

where x, y a r e the Cartesian coordinates of the physical plane, qo(x, y )

i s the velocity potential of the basic steady flow, qI (x, y , t ) i s the per -

turbation potential, being independent of E . It may be noted that in the

present formulation the space variables (x, y ) a r e not perturbed. Strict-

ly speaking, the function (x, y ) i s defined only a t points within the 0

region of the basic steady flow, whereas ~ ( x , y, t ; E ) may exist a t points

outside that region a s dictated by the perturbed flow configuration. Under

such circumstances it i s assumed that the basic flow potential vo(x,.y) -

may be continued analytically into the region wherever needed. It i s '

clear that q(x, y , t ; E ), %(x, y ) and 'P (x, y, t ) a r e all harmonic func - 1

tions of x , y. We may further introduce the complex variable z = x + iy,

the complex potential f = P + i 4 , and the complex velocity w = u - iv,

defined by:

with q denoting the velocity magnitude and 8 the flow inclination with

the positive x-axis. The coresponding expansions, of f and w a r e

f(L t ; E ) = fo(z) + € f (z , t ) + 0(t2), 1

Here fo(z) = ~ ( x , Y ) + i+o(x, y ) i s th.e complex potential of the basic flow,

f l ( z , t ) = Wl (x, y, t ) t i + (x, y, t ) i s the complex perturbation potential, both 1

being analytic functions of z.

The pressure p is given by the Bernoulli equation

where p i s the constant fluid denoit-j, C may be a function of t only,

which, after being absorbed by the t e r m B q / a t , may be taken a constant.

Consistent with the above perturbation scheme, p i s written in the form

The pressure po of the basic flow satisfies the steady form of (5).

1 1 I 1 1 1 2 - P P o + ~ ( ~ ~ o ) 2 = p p , + z u 2 = - p P c + z q c , (7 )

where pa, U a r e respectively the f ree s t ream pressure and velocity,

pc the constant cavity pressure , qc the constant flow speed on the cavity

boundary of the basic flow, which i s characterized by the cavitation num-

ber a defined by

1 0 = (pa- p c m z pu21 = (qc/ u)' - 1. (8

To facilitate the subsequent analysis, i t is convenient to introduce

a se t of intrinsic coordinate ( s , n ) a s a n alternative space variable,

where s i s the a r c length measured along a streamline in the direction

of the basic flow, and n the distance measured normal to a streamline

in the direction of increasing $ a s shown in Figure 1. Thus, the 0,

functions s(po, %), n(wot $o) can be defined by

- dvo - q0('PoY LCO)dsr d4Jo = qo('Po9 4 p n 9 ( 9 4

with ds measured along $ = constant and dn along qo = constant. 0

Consequently, the differentiations wieh respect to s and n a r e defined a s

In t e rms of (s , n), the continuity equation and ir rotationality condition be -

come respectively

The boundary conditions of this problem a r e a s follows:

(i) There a r e two boundary conditions on the f ree surface of the

cavity, one being kinematic and the other dynamic in nature. Let the I

displacement of the perturbed f ree surface of the cavity, Sf , f rom

that of the steady basic flow, Sf, be denoted by

F ( s , n , t ; ~ ) = n - ~ h ( s , t ) = 0, (11 )

so that Sf i s given by n = 0 (see Figure 1) . Then the kinematic condi-

tion that the fluid particules on the f ree surface will remain on it re-

quires

By noting the definition of s(qo, $o), n(q0, +o) given by ( 9 ) , one finds

f rom (1 1 ) that

Also f rom (9) one readily derives that

and similarly,

Substituting (1 3), (14) into (12) , and using the expansion (1 ), one obtains ,

up to the order E ,

- ah ah = - a t f 9, z t h - a s on n = ~ h ( s , t ) ,

in which use has been made of the general relationship 8 q /as = 0 qo * and

a qo/8n = 0. After expanding the quantities in (1 5) about the undisturbed

f ree surface, t>r n = 0, it is obvious that the same expression a s above

holds valid on n = 0. Now, by further applying the boundary condition

- that on n = 0, qo - qc, which is a constant, the kinematic condition

finally becomes

For the dynamic condition, we assume here that the perturbed

cavity boundary i s subject to a prescribed unsteady, but uniform

pressure perturbation, .I.

P(S, n, t ; ) = pc t E P".(t) on

Substituting (17 ) and (1 ) into the Bernoulli equation

property a %/an = 0, one finds

n = E h(s , t) .

(5), and using the

Expanding various quantities in the above equation about n = o, and using

(avola s ) = qc on n = 0, one obtains, up to the order E ,

Now, f rom the irrotationality condition (1 0b) for the basic flow,

where R i s the radius of curvature of the steady cavity boundary, the

- (or t) sign holds for the upper (or lower) branch of the cavity wall.

These signs a r e necessary to make R always a positive quantity. For

a steady cavity flow it i s a s sumed that the cavity p ressure i s a minimum

pressure in the flow field, which implies that the cavity surface of the basic

flow i s concave when viewed f rom the cavity, hence 8618 s is negative on the

upper cavity boundary A1 and positive on the lower boundary BI (cf.

Figure 1 ). The f i r s t order dynamic condition i s therefore

Equations (16 ) and (1 8 ) a r e two conditions on the cavity f ree

surface; they can be combined into one for cp by eliminating h, 1

g iving

where

At this point it i s of interest to note that if q '/R i s regarded C

a s a n equivalent gravitational acceleration g and the s -coordinate i s

rect i l inear , then (16) and (18), or equivalently (19), a r e in the same

fo rm a s those boundary' conditions i n the classical water wave problems

in a gravity field, with g pointing towards the interior of the flow. Thus,

the centrifugal acceleration q '/R due to the curvature of the basic flow C

s treamline now plays the role of the restoring force, much the same a s

gravity in water waves, in producing and propagating the surface waves

along the curved cavity boundary. I t may be remarked here that, if the

perturbed f r ee surface is approximated by i ts unperturbed steady f ree

s t r eamline boundary, thereby releasing the kinematic condition (1 6 ) and

a lso the t e r m with h in (18), then the essential restoring force i s a l -

together dropped out. On physical grounds, it may therefore be expected

that the p r e s ent formulation will yield resul t fundamentally-diff e r ent

f rom a theory using the approximation that the perturbed f ree surface

be replaced by a streamline.

The f ree surface condition (19) may also be expressed in a

complex variable form. F i r s t we note f rom (4) that on S of the f

basic flow,

Hence,

For compactnes's, we impose the normalization that q = 1. Then, by C

using (9b) and the Cauchy-Riemann equation aql/a J$ = - a Jll /aqo, (1 9)

can be finally written a s

which is the f ree surface condition in a complex variable form.

(ii) At the solid surface the normal component of the flow velo-

city relative to the moving boundary must vanish. Again using the in-

trinsic coordinates ( s , n ) , let the displacement of the wetted side of the

solid body, sot9 from i ts basic position S be prescribed by 0

over a range of s covered by the solid surface in the steady flow. Then

i t is clear that the kinematic condition (15) also holds valid-on the solid

surface So (or n = 0), that i s ,

h being here a known function of s and t. The equivalent complex

form of (22) is readily seen to be

Note that the speed qo of the basic flow on the solid surface So is

not a constant.

(iii) The condition a t the point a t infinity depends on the f ree

s t ream velocity and on whether o r not the cavity volume i s permitted to

change with time. If the f ree s t ream velocity has a prescribed small

perturbation U (t) (U may be complex), so that the free s t ream velo- 1 1

city i s U, = U t E U (t)', then we require that 1

Furthermore, when changes in the cavity volume a r e involved, then an

appropriate representation of the flow can be made by introducing a

fluid source a t the point a t infinity, a s discussed by wutl 9) . For, such

cases, a finite cavity model for the basic flow i s required to incorporate

the source a t infinity into the flow problem, a s shown previously by the

present authors(20). These points have been further justified from a

physical and mathematical ground by I3 enjamid2 ). Furthermore , from

Kelvin's theorem on the conservation of circulation, the circulation

around the point a t infinity cannot be changed in unsteady flow for

t < oo. Therefore, in addition to condition (24), it i s required that

where r i s a contour around the point a t infinity and Q (t) i s equal 1

to the time ra te of increase in cavity volume, which i s supposed to be

prescribed.

Finally, we state that if the problem i s of the initial value type,

then .no radiation condition i s needed for the surface waves; these waves

will turn out to propagate automatically towards downstream. However,

in the case of simple harmonic motions, when being treated a s a quasi-

steady flow, then the so-called "radiation condition" will be needed to

ensure that the waves generated by body motion will not propagate up-

s t ream on the cavity surface. This completes our formulation of the

problem.

SURFACE WAVES ON A HOLLOW VORTEX

This relatively simple problem was chosen to demonstrate an

application of this general theory to a special case; this application i s

partly meant for a verification of the complicated expression of the

f ree surface boundary condition, since the problem has already been

solved by Lord Kelvin (I8) in a completely different way.

The basic flow i s an irrotational, circulating motion about a

point, say z = 0, a s center. The f ree surface will be deaoted by

/ z 1 = a on which the speed of flow i s normalized to unity. The velocity

potential i s simply

and therefore,

The transformation

fJz) = - i a log z

maps the entire basic flow in the hodograph w -plane, two 1 ,< 1, into 0

the upper half of the 5-plane (see figure 2 ) .

We assume that the cavity pressure be kept a t constant, that is,

pl(t) = 0, then the boundary conditions of this problem a re

a a Z d 1 dWo a a 1 where L = (;3f+E) - (-log (-

dWo a - ) l ( r + n l - ~ - - 0 dfo "0 dfo 0 0 dfo df '

0

and

where I' i s a contour around the point a t infinity, or,

solution of the above boundary value problem is

L(f ) =O. 1

The complementary solution can be written as

(31)

A particular

where c (t) a r e unknown functions of t. In order to satisfy equations n

(30)and(31), that is , L( f l )= O [ ( G -i)"], , > 0 , a s i and l ~ ( f ) / < m 1

a t every point on the free surface it is necessary that c = 0 for a l l n. n

Therefore, the solution of this boundary value problem is given by (32)

alone.

By use of equations (26) and (27), equation (32) can be changed

into

Since f (z , t ) should be regular everywhere outside lzl = a, we may 1

write

1 f $ z , t ) = A n ( t ) y .

z n = 1, 2,...

Substitution of equation (35 ) into equation (34) gives

where dot represents the differentiation with respect to t. The solution

of the last equation is

where a a r e constants. Finally, we have n

where the constants a and p n can be determined by appropriate n

initial conditions. The above result agrees with that obtained by

(18) LordKelvin .

ACKNOWLEDGMENT

This work i s sponsored in par t by the Office of Naval Research

of the U. S. Navy, under contract Nonr -22O(35). Reproduction in

whole or in par t is permitted for any purpose of the United States

Government.

BIB LIOGRAPHY

ijber dis continuier liche Fliis sigkeitsbewegungen. Helmholtz, H., Monatsber. Berlin. Akad. 1868, pp. 2 15-228.

Zur Theorie f r e i e r F lus sigkeitsstrahlen. Kirchhoff, G., J. reine angew. Math. Vol. 70, 1869, pp. 289-298.

J e t s and Cavities; Encyclopedia of Physics, Vol. IX. Gilbarg, D. , Springer -Verlag, Berlin, Gottingen, Heidelberg, 1960, pp. 356- 363; pp. 321 -326.

J e t s , Wakes and Cavities. Birkhoff, G. & Zarantonello, E. H., Academic P r e s s , New York, 1957, pp. 236-257.

Collected Works of Theodore von ~ & r m 6 n , Vol. 4. von ~ 6 r m A n , T. , Betterworth Scientific Publications, London, 1956, pp. 396 -398.

Unsteady Flow with F r e e Boundaries. Gilbarg, D. , Zeitschrift fiir angewandte Mathematic und Physik, Vol. 3, 1952, pp. 34-42.

Unsteady Cavitating Flow P a s t Curved Obstacles. Woods, L. C., A. R. C. Technical Report C. P. No. 149, 1954.

Finite Two-Dimensional Cavities. Yih, C. S. , Proc. Roy. Soc. , London. Series A, Vol. 256, 1960, pp. 90-100.

Perturbat ion of F r e e Surface Flows. Ablow, C. M. and Hayes, W. D. , Tech. Report 1, Oivision Applied Math. , Brown Univ. , 1951.

On the Stability of Some Flows of a n Ideal Fluid with F r e e Surfaces. Fox, J. L. and Morgan, G. W., Quarter ly of Applied Math., Vol. 11, 1954, pp. 439-456.

Unsteady Two-Dimensional Flows with F r e e Boundaries. Curle, N. , Proc. Roy. Soc., London. Series A, Vol. 235, 1956, pp. 375-395.

Unsteady Plane Flow P a s t Curved Obstacles with Infinite Wakes. Woods, L. C. , Proc. Roy. Soc., London. Ser ies A, Vol. 229, 1955, pp. 152-180.

Fully Cavitating Hydrofoils i n Nonsteady Motion. Parkin, B. R. , Engineering Division Report No. 85-2, Calif. Inst. of Tech. , 1957.

A Linearized Theory for Nonsteady Cavity Flows. Wu, T. Y . , Engineering Division Report No. 85-6, Calif. Inst. of Tech., 1957.

A General Linearized Theory for Cavitating Hydrofoils i n Non- steady Flow. Timman, R. , Second Symposium, Naval Hydrodynamics, Washington, D. C. , U. S. A. , 1958, pp. 559-579.

Unsteady Motion of A Flat Plate i n A Cavity Flow. Geurst, J. A. , Report No. 21, Inst. of Applied Math. , Technological Univ. , Delft, Holland, 1959.

BIB LIOGRAPHY (continued)

17. ' A Linearized Theory for the Unsteady Motion of a Supercavitating Hydrofoil. Geurst, J. A. , Report No. 22, Inst. of Applied Math. , Technological Univ. , Delft, Holland, 1 96 0.

18. Vibration of a Columnar Vortex. Thompson, Wm. , Philosophical Magazine, 1 O(5), 1880, pp. 155-168.

19. Unsteady Supercavitating Flows. Wu, T. Ye , Second Symposium, Naval Hydrodynamics, Washington, D. C., U. S.A., 1958, pp. 293-313.

20. Small Time Behavior of Unsteady Cavity Flows. Wang, D. P. and Wu, T. Ye , Archive for Rational Mechanics and Analysis, Vol. 14, No. 2, 1963, pp. 127-152.

2 1. Note on the interpretation of two-dimensional theories of growing cavities, Benjamin, T. B. , Journal of Fluid Mechanics, Vol. 19, 1964, pp. 137-144.

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(Department of Naval Archi tecture) Professor W. W. Willmarth (Department of Aeronautical Engineering)

Dr. L. G. Straub, Director St. Anthony Fa l l s Hydraulic Laboratory University of Minnesota Minneapolis 14, Minnesota Attn: Mr. J. N. Wetzel

Professor B. Silberman

Professor J. J. Foody Engineering Department New York State University Mari t ime College F o r t Schylyer, New York

New Y ork University Institute of Mathematical Sciences 25 Waverly P lace New York 3, New York Attn: Professor J. Keller

Professor J. J. Stoker

The Johns Hopkins University Department of Mechanical Engineering Baltimore 18, Maryland Attn: Professor S. Corrs in

Professor 0. M. Phillips (2)

Massachusetts Institute of Technology Department of Naval ,Architecture and Marine Engineering Cambridge 39, Massachusetts Attn: Professor M. A. Abkowitz

Dr. G. F. Wislicenus Ordnance Research Laboratory Pennsylvania State University University Park , Pennsylvania Attn: Dr. M. Sevik

Professor R. C. DiPrima Department of Mathematics Rensselaer Polytechnic Institute Troy, New York -

Director Woods Hole Oceanographic Institute Woods Hole, Massachusetts

Stevens Institute of Technology Davids on Laboratory Castle Point Station Hoboken, New J e r s e y Attn: Mr. D. Savitsky

Mr. J. P. Bres l in Mr. C. J. Henry Mr. S. Tsakonas

Webb Institute of Naval ~ r c h i t e c t u r e Crescent Beach Road Glen Cove, New York Attn: Professor E. V. Lewis

Technical Library

Executive Director Air F o r c e Office of Scientific Research Washington 25, D. C. Attn: Mechanics Branch

Commander Wright Ai r Development Division Aircraf t Laboratory Wright-Patterson Air F o r c e Base, Qhio Attn: Mr. W. Mykytow,

Dynamics Branch

Cornell Aeronautical Laboratory 4455 Genesee Street Buffalo, New York Attn: Mr. W. Targoff

Mr. R. White

Massachusetts Institute of Technology Fluid Dynamics Research Laboratory Cambridge 39, Massachusetts Attn: Professor H. Ashley

Professor M. Landahl Professor J. Dugundji

Hamburgische Schiffbau-Versuchsanstalt Brarnfelder S t rasse 164 Hamburg 33, Germany Attn: Dr. H. Schwanecke

Dr. H. W. Lerbs

Institut fu r Schiffbau der Univer s i ta t Hamburg Berl iner Tor 21 Hamburg 1, Germany Attn: Professor G. P. Weinblum

Transportation Technical Research Institute P -1 057, Mejiro-Cho, Toshima-Ku Tokyo, Japan

Max-Planck Institut f u r Stromung s fors - chung Bott ingerstrasse 618 Gottingen, Germany Attn: Dr. H. Reichardt

Hydro-og Aerodynamisk Laboratorium Lyngby, Denmark Attn: Professor Ca r l Prohaska

Skip smodelltanken Trondheim, Norway Attn: P ro fe s so r J. K. Lunde

Ver suchsanstalt fu r Was serbau and Schiffbau Schleuseninsel i m Tiergar ten Berl in , Germany Attn: Dr. S. Schuster, Director

Dr. Grosse

Technische Hogeschool Institut voor Toegepaste Wiskunde Julianalaan 132 Delft, Netherlands Attn: P ro fe s so r R. Timman

Bureau D1Analyse e t de Recherche Applique e s 47 Avenue Victor Bresson Issy-Les -Moulineaux Seine, F rance Attn: Professor Siestrunck

Netherlands Ship Model Basin Wageningen, The Netherlands Attn: Dr. Ir. J. D. van Manen

National Physical Laboratory Teddington, Middlesex, England Attn: Mr. A. Silverleaf,

Superintendent Ship Division Head, Aerodynamics Division

Head, Aerodynamics Department Royal Air c ra f t Establishment Farnborough, Hants, England Attn: Mr. M. 0. W. Wolfe

Dr. S. F. Hoerner 148 Busteed Drive Midland Pa rk , New ~ e r s e ~

Boeing Airplane Company Seattle Division Seattle, Washington Attn: Mr. M. J. Turner

Elec t r ic Boat Division General Dynamics Corporation Groton, Connecticut Attn: Mr. Robert McCandliss

General Applied Sciences Labs. , Inc. Merr ick and Stewart Avenues Westbury, Long Island, New York

Gibbs and Cox, Inc. 21 West Street New Ysrk, New York

Lockheed Aircraf t Corporation Missi les and Space Division Pa lo Alto, California Attn: R. W. Kermeen

Grumman Aircraf t Engineering Corp. Bethpage, Long Island, New York Attn: Mr. E. Bai rd

Mr. E. Bower Mr. W. P. Car l

Midwest Research Institute 425 Volker Blvd. Kansas City 10 Missouri Attn: Mr. Zeydel

Director , Department of Mechanical Sciences Southwest Re sea rch Institute 8500 Culebra Road San Antonio 6, Texas Attn: Dr. H. N. Abramson

Mr. G. Ransleben Editor , Applied Mechanics Review

Convair A Division of General Dynamics San Diego, California Attn: Mr. R. H. Oversmith

Mr. H. T. Brooke

Hughes Tool Company Air c raf t Division Culver City, California Attn: Mr. M. S. Harned

Hydronautics, Incorporated Pindell School Road Howar d County Laurel , Maryland Attn: Mr. Phill ip Eisenberg

Rand Development Corporation 13600 Deise Avenue Cleveland 10, Ohio Attn: Dr. A. S. Iberal l

AVCO Corporation Lycoming Division 1701 K Street , N. W. Apt. No. 904 Washington, D. C. Attn: Mr. T. A. Duncan

Mr. J. G. Baker Baker Manufacturing Company Evansville, Wisconsin

Cur t i ss -Wright Corporation Research Division Turbomachinery ~ i v i s i d n Quehanna, Pennsylvania Attn: Mr. George H. Pedersen

Dr. Blaine R. Parkin AiResearch Manufacturing Corporation 9851 -995 1 Sepulveda Boulevard Los Angeles 45, California

The Boeing Company Aero-Space Division Seattle 24, Washington Attn: Mr. R. E. Bateman

Internal Mail Station 46 -74

Lockheed Aircraf t Corporation California Division Hydrodynamics Research Burbank, California Attn: Mr. Bill Eas t

National Research Council Montreal Road Ottawa 2, Canada Attn: Mr. E. S. Turner

The Rand Corporation 1700 Main Street Santa Monica, California Attn: Technical Library

Stanford University Department of Civil Engineering Stanford, California Attn: Dr. Byrne P e r r y

U. S. Rubber Company Dr. E. Y. Hsu Research and eve-lopment Department Wayne, New J e r s e y Attn: Mr. L. M. White

Technical Research G r ~ u p , Inc. Route 110 Melville, New York, 11749 Attn: Mr. J ack Kotik

Mr. C. w i i l e y - - F l a t 102 6 -9 Charterhouse Square London, E. C. 1, England

Dr. Hir sh Cohen IBM Research Center P. 0. Box 218 Yorktown Heights, New York

Mr. David Wellinger Hydrofoil Pro jec ts Radio Corporation of America Burlington, Massachusetts

Food Machinery Corporation P. 0. Box 367 San Jose , California Attn: Mr. G. Tedrew

Dr. T. R. Goodman Oceanics , Inc. Technical Industrial Park Plainview, Long Island, New York

Prof es sor Brunelle Department of Aeronautical Engineering Princeton, New Je r sey

Commanding Officer Office of Naval Research Branch Office 230 N. Michigan Avenue, Chicago 1, Illinois

University of Colorado Aerospance Engineering Sciences Boulder, Colorado Attn: Professor M. S. Uberoi

The Pennsylvania State University Department of Aeronautical Engineering Ordnance Research Laboratory P. 0. Box 30 State College, Pennsylvania Attn: Pr ofes sor J. William Holl

Institut fur Schiffbau der Univer sitat Hamburg Lammer sieth 9 0 2 Hamburg 33, Germany Attn: Dr. 0. Grim

Technische Hogeschool Laboratorium voor Scheepsbounkunde Mekelweg 2, Delft, The Netherlands Attn: Professor Ir. J. Gerri tsma